The ball-on-chain method is, of course, what is known in the Chinese internal styles as fajin 發勁. (Balgeong 발경 in Korean.) It is described by one martial artist as “impulse” or the “explosive transfer of momentum” and by another as “impact.”
So which is it for ITF Taekwon-Do, crowbar or ball-on-chain?
Well, let's look at refer to the ITF Encyclopaedia in search of the answer. In the section on Concentration in the Theory of Power we find the following statement: “the shorter the time for the concentration, the greater will be the power of the blow” (Vol. 2, p. 20). Another section in the Encyclopaedia concerning attacking techniques states the following: “The moment the attacking tool reaches the target, pull it back . . .” (Vol. 3, p. 17). The attacking tool does not spend unnecessary time on the target. Once the momentum is transferred, it is pulled-back. Regarding punching the Encyclopaedia says that one should: “Avoid unnecessary tension of the arms and shoulders” and “Relax the muscles immediately after the fist has reached the target” (Vol. 3, p. 29).
In other words, the attacking tool acts more like a ball-on-chain, very quickly transferring its momentum into the target and immediately “relaxing” afterwards. The structure is not kept like a crowbar, but instead it relaxes like a ball-on-chain. Techniques in ITF Taekwon-Do also adhere to the principle of kinetic chaining where the “hip is jerked slightly before the action in order to concentrate the larger muscles of the hip and abdomen together with the smaller muscles of the four extremities against the target simultaneously.” This, of course happens sequentially. For example, in a punch, first the hips rotates towards the target, then the shoulders, then the arm is snapped forward—all of this is preceded by motion from the legs based on the “knee spring”—forming one continuous kinetic chain.
If fajin / balgyeong can be defined as kinetic chaining with emphasis on the quick transferal of momentum—impulse or impact—into the target, achieved by relaxed movements before and directly after the blow, then I believe ITF Taekwon-Do more closely follow the ball-on-chain method than the crowbar method. In other words, an ITF Taekwon-Do punch is more like a Hsing-I Quan punch than a Karate punch.
It is important to remember that this has not always been (and in a sense is still not always) the case. Taekwon-Do's father is Shotokan Karate and in the early days of Taekwon-Do the movements very much resembled Karate. The real change towards this balgyeong way of moving occurred later in ITF Taekwon-Do's evolution (I'm guessing the early 80s) even though the principles (i.e. Theory of Power) were set to paper from quite early on.
However, the observant practitioner will discover that ITF Taekwon-Do does not utilise the balgyeong method exclusively. ITF Taekwon-Do seems to be using both the ball-on-chain and the crowbar, depending on the technique. The turning kick, for instance, works on the ball-on-chain principle, while the spinning reverse turning kick (not the reverse hook kick) is based on the crowbar principle. The basic front punch employs the ball-on-chain method while the ridge-hand strike uses the crowbar method. The front snap kick is ball-on-chain; the front pushing kick is crowbar. The twisting kick, ball-on-chain; the downward kick, crowbar. And the side-piercing kick can be performed either on the ball-on-chain principle or on the crowbar principle, depending on the desired effect.
Although there has been an evolution in how ITF Taekwon-Do approaches power generation, it has not completely thrown off its Shotokan Karate heritage. It would seem that power generation in ITF Taekwon-Do is situational; sometimes used typically Karate ways of power generation, other times using balgyeong. The determining factor is usually the technique employed, but could also be the effect desired. What is significant, however, is that some of the most iconic Karate-like movements, for instance the walking stance front fore fist punch or the low forearm block, doesn't strictly employ the same Karate-like mechanics any more.